3.152 \(\int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx\)

Optimal. Leaf size=87 \[ \frac{1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 c^2 x^3+\frac{2}{9} b d x^9 (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{11} b^2 d^2 x^{11} \]

[Out]

(a^2*c^2*x^3)/3 + (2*a*c*(b*c + a*d)*x^5)/5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x
^7)/7 + (2*b*d*(b*c + a*d)*x^9)/9 + (b^2*d^2*x^11)/11

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Rubi [A]  time = 0.1741, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 c^2 x^3+\frac{2}{9} b d x^9 (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{11} b^2 d^2 x^{11} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^2)^2*(c + d*x^2)^2,x]

[Out]

(a^2*c^2*x^3)/3 + (2*a*c*(b*c + a*d)*x^5)/5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x
^7)/7 + (2*b*d*(b*c + a*d)*x^9)/9 + (b^2*d^2*x^11)/11

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Rubi in Sympy [A]  time = 28.6622, size = 87, normalized size = 1. \[ \frac{a^{2} c^{2} x^{3}}{3} + \frac{2 a c x^{5} \left (a d + b c\right )}{5} + \frac{b^{2} d^{2} x^{11}}{11} + \frac{2 b d x^{9} \left (a d + b c\right )}{9} + x^{7} \left (\frac{a^{2} d^{2}}{7} + \frac{4 a b c d}{7} + \frac{b^{2} c^{2}}{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x**2+a)**2*(d*x**2+c)**2,x)

[Out]

a**2*c**2*x**3/3 + 2*a*c*x**5*(a*d + b*c)/5 + b**2*d**2*x**11/11 + 2*b*d*x**9*(a
*d + b*c)/9 + x**7*(a**2*d**2/7 + 4*a*b*c*d/7 + b**2*c**2/7)

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Mathematica [A]  time = 0.0293511, size = 87, normalized size = 1. \[ \frac{1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 c^2 x^3+\frac{2}{9} b d x^9 (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{11} b^2 d^2 x^{11} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x^2)^2*(c + d*x^2)^2,x]

[Out]

(a^2*c^2*x^3)/3 + (2*a*c*(b*c + a*d)*x^5)/5 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x
^7)/7 + (2*b*d*(b*c + a*d)*x^9)/9 + (b^2*d^2*x^11)/11

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Maple [A]  time = 0., size = 90, normalized size = 1. \[{\frac{{b}^{2}{d}^{2}{x}^{11}}{11}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{2}{d}^{2}+4\,cabd+{b}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){x}^{5}}{5}}+{\frac{{a}^{2}{c}^{2}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+a)^2*(d*x^2+c)^2,x)

[Out]

1/11*b^2*d^2*x^11+1/9*(2*a*b*d^2+2*b^2*c*d)*x^9+1/7*(a^2*d^2+4*a*b*c*d+b^2*c^2)*
x^7+1/5*(2*a^2*c*d+2*a*b*c^2)*x^5+1/3*a^2*c^2*x^3

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Maxima [A]  time = 1.3458, size = 115, normalized size = 1.32 \[ \frac{1}{11} \, b^{2} d^{2} x^{11} + \frac{2}{9} \,{\left (b^{2} c d + a b d^{2}\right )} x^{9} + \frac{1}{7} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{7} + \frac{1}{3} \, a^{2} c^{2} x^{3} + \frac{2}{5} \,{\left (a b c^{2} + a^{2} c d\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^2*x^2,x, algorithm="maxima")

[Out]

1/11*b^2*d^2*x^11 + 2/9*(b^2*c*d + a*b*d^2)*x^9 + 1/7*(b^2*c^2 + 4*a*b*c*d + a^2
*d^2)*x^7 + 1/3*a^2*c^2*x^3 + 2/5*(a*b*c^2 + a^2*c*d)*x^5

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Fricas [A]  time = 0.206938, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} d^{2} b^{2} + \frac{2}{9} x^{9} d c b^{2} + \frac{2}{9} x^{9} d^{2} b a + \frac{1}{7} x^{7} c^{2} b^{2} + \frac{4}{7} x^{7} d c b a + \frac{1}{7} x^{7} d^{2} a^{2} + \frac{2}{5} x^{5} c^{2} b a + \frac{2}{5} x^{5} d c a^{2} + \frac{1}{3} x^{3} c^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^2*x^2,x, algorithm="fricas")

[Out]

1/11*x^11*d^2*b^2 + 2/9*x^9*d*c*b^2 + 2/9*x^9*d^2*b*a + 1/7*x^7*c^2*b^2 + 4/7*x^
7*d*c*b*a + 1/7*x^7*d^2*a^2 + 2/5*x^5*c^2*b*a + 2/5*x^5*d*c*a^2 + 1/3*x^3*c^2*a^
2

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Sympy [A]  time = 0.149637, size = 100, normalized size = 1.15 \[ \frac{a^{2} c^{2} x^{3}}{3} + \frac{b^{2} d^{2} x^{11}}{11} + x^{9} \left (\frac{2 a b d^{2}}{9} + \frac{2 b^{2} c d}{9}\right ) + x^{7} \left (\frac{a^{2} d^{2}}{7} + \frac{4 a b c d}{7} + \frac{b^{2} c^{2}}{7}\right ) + x^{5} \left (\frac{2 a^{2} c d}{5} + \frac{2 a b c^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x**2+a)**2*(d*x**2+c)**2,x)

[Out]

a**2*c**2*x**3/3 + b**2*d**2*x**11/11 + x**9*(2*a*b*d**2/9 + 2*b**2*c*d/9) + x**
7*(a**2*d**2/7 + 4*a*b*c*d/7 + b**2*c**2/7) + x**5*(2*a**2*c*d/5 + 2*a*b*c**2/5)

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GIAC/XCAS [A]  time = 0.222938, size = 127, normalized size = 1.46 \[ \frac{1}{11} \, b^{2} d^{2} x^{11} + \frac{2}{9} \, b^{2} c d x^{9} + \frac{2}{9} \, a b d^{2} x^{9} + \frac{1}{7} \, b^{2} c^{2} x^{7} + \frac{4}{7} \, a b c d x^{7} + \frac{1}{7} \, a^{2} d^{2} x^{7} + \frac{2}{5} \, a b c^{2} x^{5} + \frac{2}{5} \, a^{2} c d x^{5} + \frac{1}{3} \, a^{2} c^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^2*x^2,x, algorithm="giac")

[Out]

1/11*b^2*d^2*x^11 + 2/9*b^2*c*d*x^9 + 2/9*a*b*d^2*x^9 + 1/7*b^2*c^2*x^7 + 4/7*a*
b*c*d*x^7 + 1/7*a^2*d^2*x^7 + 2/5*a*b*c^2*x^5 + 2/5*a^2*c*d*x^5 + 1/3*a^2*c^2*x^
3